Greatest common divisors of shifted primes and Fibonacci numbers

Abstract

Let $$(F_n)$$ ( F n ) be the sequence of Fibonacci numbers and, for each positive integer k, let $${\mathcal {P}}_k$$ P k be the set of primes p such that $$\gcd (p - 1, F_{p - 1}) = k$$ gcd ( p - 1 , F p - 1 ) = k . We prove that the relative density $$\varvec{r}({\mathcal {P}}_k)$$ r ( P k ) of $${\mathcal {P}}_k$$ P k exists, and we give a formula for $$\varvec{r}({\mathcal {P}}_k)$$ r ( P k ) in terms of an absolutely convergent series. Furthermore, we give an effective criterion to establish if a given k satisfies $$\varvec{r}({\mathcal {P}}_k) > 0$$ r ( P k ) > 0 , and we provide upper and lower bounds for the counting function of the set of such k’s. As an application of our results, we give a new proof of a lower bound for the counting function of the set of integers of the form $$\gcd (n, F_n)$$ gcd ( n , F n ) , for some positive integer n. Our proof is more elementary than the previous one given by Leonetti and Sanna, which relies on a result of Cubre and Rouse.

On the greatest common divisor ofnand thenth Fibonacci number, II

“…Their approach relied on a result of Cubre and Rouse [6], which in turn follows from Galois theory and the Chebotarev density theorem. Later, Jha and Sanna [10,Proposition 1.4] obtained an elementary proof as an application of related arithmetic problem over shifted primes. Leonetti and Sanna [12] also gave the upper bound #A(x) = o(x) as x → +∞, and asked for a more precise estimate.…”